DOCSLIB.ORG

Strengthened Moser’S Conjecture, Geometry of Grunsky

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Strengthened Moser’S Conjecture, Geometry of Grunsky DOI: 10.2478/s11533-007-0013-5 Research article CEJM 5(3) 2007 551–580 Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues∗ Samuel Krushkal Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA Received 31 July 2006 ; accepted 3 April, 2007 Abstract: The Grunsky and Teichmuller¨ norms κ(f)andk(f) of a holomorphic univalent function f in a finitely connected domain D ∞with quasiconformal extension to C are related by κ(f) ≤ k(f). In 1985, Jurgen¨ Moser conjectured that any univalent function in the disk Δ∗ = {z : |z| > 1} can be approximated locally uniformly by functions with κ(f) <k(f). This conjecture has been recently proved by R. Kuhnau¨ and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kuhnau¨ problem on an exact lower bound in the inverse inequality estimating k(f)byκ(f), and in the related Ahlfors inequality. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: quasiconformal, univalent function, Grunsky coefficient inequalities, universal Teichmuller¨ space, subharmonic function, Strebel’s point, Kobayashi metric, generalized Gaussian curvature, holomorphic curvature, Fredholm eigenvalues. MSC (2000): 30C35, 30C62, 32G15,30F60, 32F45, 53A35 1 Introduction and main results 1.1 Grunsky inequalities and Moser’s conjecture The classical Grunsky theorem states that a holomorphic function f(z)=z+const +O(z−1) in a neighborhood U0 of z = ∞ can be extended to a univalent holomorphic function on ∗ To Reiner Kuhnau¨ on his 70th birthday 552 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 the disk Δ∗ = {z ∈ C = C ∪{∞}: |z| > 1} if and only if its Grunsky coefficients αmn satisfy the inequalities ∞ √ mn αmnxmxn ≤ 1, (1.1) m,n=1 where αmn are defined by ∞ f(z) − f(ζ) log = − α z−mζ−n, (z, ζ) ∈ (Δ∗)2, (1.2) z − ζ mn m,n=1 ∞ 2 2 2 2 x =(xn) runs over the unit sphere S(l )oftheHilbertspacel with x = |xn| ,and 1 the principal branch of the logarithmic function is chosen (cf. [1]). The quantity ∞ √ 2 κ(f):=sup mn αmnxmxn : x =(xn) ∈ S(l ) (1.3) m,n=1 is called the Grunsky constant (or Grunsky norm)off. Let Σ denote the collection of all univalent holomorphic functions −1 ∗ f(z)=z + b0 + b1z + ···:Δ → C \{0}, (1.4) and let Σ(k) be its subset of the functions withk-quasiconformal extensions to the unit {| | } 0 disk Δ = z < 1 so that f(0) = 0. Put Σ = k Σ(k). This collection closely relates to universal Teichmuller¨ space T modelled as a bounded domain in the Banach space B of holomorphic functions in Δ∗ with norm 2 2 ϕ B =sup(|z| − 1) |ϕ(z)|. (1.5) Δ∗ All ϕ ∈ B can be regarded as the Schwarzian derivatives 2 Sf =(f /f ) − (f /f ) /2 of locally univalent holomorphic functions in Δ∗.ThepointsofT represent the functions f ∈ Σ0 whose minimal dilatation μ μ ∗ k(f):=inf{k(w )= μ ∞ : w |∂Δ = f} determines the Teichmuller¨ metric on T.Herewμ denotes a quasiconformal homeomor- phism of C with the Beltrami coefficient μ(z)=∂zw/∂zw, (1.6) and μ ∞ = ess supC |μ(z)| < 1. (1.7) S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 553 Grunsky’s theorem was strengthened for the functions with quasiconformal extensions by several authors. The basic results obtained by Kuhnau,¨ Pommerenke and Zhuravlev are as follows: for any f ∈ Σ0, we have the inequality κ(f) ≤ k(f); (1.8) on the other hand, if a function f ∈ Σ satisfies the inequality κ(f) <kwith some constant k<1, then f has a quasiconformal extension to C with a dilatation k1 = k1(k) ≥ k (see [2–4]and[5], pp. 82–84). An explicit bound k1(k) is given in [6] (see Section 5). A crucial point is that for a generic function f ∈ Σ0,wehavein(1.8) the strict inequality κ(f) <k(f) (1.9) (see, e.g., [7–10]). The functions, for which equality holds in (1.8), play an important role in applications of the Grunsky inequality technique. A characterization of such functions is presented below. In 1985, Jurgen¨ Moser conjectured that the set of functions with κ(f)=k(f) (1.10) is rather sparse in Σ0 so that any function f ∈ Σ is approximated by functions satisfying (1.9) uniformly on compact sets in Δ∗. This conjecture was recently proved in [8]. A related conjecture, posed in [8] and still remaining open, states that f ∈ Σ0 sat- isfying (1.9) cannot be the limit functions of locally uniformly convergent sequences 0 {fn}⊂Σ with κ(fn)=k(fn). This was proved in [11] under the assumption that approximating maps fn are asymp- totically conformal on the unit circle S1 = ∂Δ∗. 1.2 Main theorem Uniform convergence on compact sets is natural and sufficient in many problems of ge- ometric complex analysis. In applications of Schwarzian derivatives, especially to Teich- muller¨ spaces, one has to use the strong topology defined by the norm (1.5). A question is: How sparse is the set of derivatives ϕ = Sf in T representing the maps with the property (1.10)? Our first main result answers this question strengthening Moser’s conjecture. 0 Theorem 1.1. The set of points ϕ = Sf , which represent the maps f ∈ Σ with κ(f) <k(f), is open and dense in the space T. Openness follows from continuity of both quantities κ(f)andk(f) as functions of the Schwarzian derivatives Sf on T (cf. [12]). The main part of the proof concerns the density. The proof involves the density of Strebel points in T and relies on curvature properties of certain Finsler metrics on this space. 554 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 1.3 Application to Fredholm eigenvalues The Fredholm eigenvalues ρn of a smooth closed Jordan curve L ⊂ C are the eigenvalues of its double-layer potential, i.e., of the integral equation ρ ∂ 1 u(z)+ u(ζ) log dsζ = h(z), π ∂nζ |ζ − z| L which often appears in applications (see e.g. [5, 13–18] and the references cited there). These values are intrinsically connected with the Grunsky coefficients of the corre- sponding conformal maps. This is qualitatively expressed by the remarkable Kuhnau-¨ Schiffer theorem on reciprocity of κ(f) to the least positive Fredholm eigenvalue ρL.It is defined for any oriented closed Jordan curve L ⊂ C by 1 |D (u) −D ∗ (u)| =sup G G , ρL DG(u)+DG∗ (u) where G and G∗ are, respectively, the interior and exterior of L; D denotes the Dirichlet integral, and the supremum is taken over all functions u continuous on C and harmonic on G ∪ G∗ (cf. [10, 16]). In general, there is only a rough estimate for ρL by Ahlfors’ inequality 1 ≤ qL, (1.11) ρL where qL is the minimal dilatation of quasiconformal reflections across L, (that is, of the orientation reversing quasiconformal homeomorphisms of C preserving L point-wise); see, e.g., [15, 19, 20]. As a consequence of Theorem 1.1,wehave Theorem 1.2. The set of quasiconformal curves L, for which Ahlfors’ inequality (1.11) is satisfied in the strict form 1/ρL <qL, is open and dense in the strongest topology determined by the norm (1.5). Proof. Since all quantities in (1.11) are invariant under the action of the M¨obius group PSL(2, C)/ ± 1, it suffices to exploit quasiconformal homeomorphisms f of the sphere C 1 ∗ carrying the unit circle S = ∂Δ onto L whose Beltrami coefficients μf (z)=∂z¯f/∂zf have support in the unit disk Δ and which are normalized via (1.4), i.e., with restrictions ∗ 0 μ f|Δ ∈ Σ . Then the reflection coefficient qL equals the minimal dilatation k(w )= μ ∞ of quasiconformal extensions wμ of f|Δ∗ to C,andTheorem1.2 immediately follows from Theorem 1.1. In the last section we solve Kuhnau’s¨ problem related to comparing the Teichmuller¨ and Grunsky norms and establish the sharp lower bound in the inverse inequalities esti- mating k(f)byκ(f)orρL. This important problem has been open since 1981. S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 555 2 Preliminaries We briefly present here certain underlying results needed for the proof of Theorem 1.1. The exposition is adapted to our special case. 2.1 Frame maps and Strebel points For a map f μ ∈ Σ0,wedenoteby[f μ] its equivalence class containing the maps f ν ∈ Σ0 which coincide with f μ on the unit circle S1 (and hence on Δ∗). μ0 Let f0 := f be an extremal representative of its class [f0] with dilatation μ μ 1 1 k(f0)= μ0 ∞ =inf{k(f ):f |S = f0|S } = k, and assume that there exists in this class a quasiconformal map f1 whose Beltrami coef- ficient μf1 satisfies the strong inequality | | ess supAr μf1 (z) <k in some annulus Ar := {z : r<|z| < 1}.Thenf1 is called a frame map for the class [f0] and the corresponding point of the space T is called a Strebel point. The following two results are fundamental in the theory of extremal quasiconformal maps and Teichmuller¨ spaces. Proposition 2.1. (cf. [21]) If a class [f] has a frame map, then the extremal map f0 in this class is unique and either conformal or a Teichmuller¨ map with Beltrami coefficient μ0 = k|ψ0|/ψ0 on Δ, defined by an integrable holomorphic quadratic differential ψ on Δ and a constant k ∈ (0, 1).
Load more

Recommended publications
  • A Study on Concave Functions in Geometric Function Theory
    TOHOKU UNIVERSITY Graduate School of Information Sciences A Study on Concave Functions in Geometric Function Theory Dissertation written by: Rintaro Ohno born 21. January 1984 in Tokyo (Japan) Sendai 2014 Division of Mathematics Prof. Toshiyuki Sugawa ii Supervisor: Prof. Toshiyuki Sugawa Examiners: Prof. Shigeru Sakaguchi Prof. Reika Fukuizumi Date of submission: 4. August 2014 Contents Introduction iv Notations vii 1 Basic Properties and Preliminaries1 1.1 Univalent Functions and Basic Principles......................1 1.2 Convex Functions....................................3 2 Concave Functions5 2.1 Characterizations for Concave Functions.......................6 2.2 Integral Representations for Concave Functions................... 12 3 Coefficients of Concave Functions - Known Results 18 3.1 Coefficients of the Taylor Series............................ 18 3.2 Coefficients of the Laurent Series........................... 24 3.3 Alternative Proof for the Residue........................... 33 4 Extension of Necessary and Sufficient Conditions for Concave Functions 35 4.1 Proofs of the Extended Formluas........................... 36 4.2 Application of the Extended Condition........................ 39 5 On a Coefficient Body of Concave Functions 42 5.1 Representation Formula for Cop and Lemmas.................... 44 5.2 Proof of the Theorems................................. 49 Acknowledgments 54 iii Introduction In the course of the last century, the field of geometric function theory presented many interesting and fascinating facts. Starting with the mapping theorem of Riemann, Bieberbach [5] gave a long-lasting conjecture in 1916, which attracted the attention of many mathematicians over the time. The conjecture concerned the class S of analytic and univalent functions in the unit disk 0 D = fz 2 C : jzj < 1g, normalized at the origin to have f(0) = f (0)−1 = 0, and stated that the P1 n Taylor coefficients an(f) of these functions f(z) = z + n=0 an(f)z 2 S satisfy the inequality jan(f)j ≤ n.
    [Show full text]
  • Arxiv:1911.03589V1 [Math.CV] 9 Nov 2019 1 Hoe 1 Result
    A NOTE ON THE SCHWARZ LEMMA FOR HARMONIC FUNCTIONS MAREK SVETLIK Abstract. In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given. 1. Introduction 1.1. A summary of some results. In this paper we consider some generalizations of the Schwarz lemma for harmonic functions from the unit disk U = {z ∈ C : |z| < 1} to the interval (−1, 1) (or to itself). First, we cite a theorem which is known as the Schwarz lemma for harmonic functions and is considered as a classical result. Theorem 1 ([10],[9, p.77]). Let f : U → U be a harmonic function such that f(0) = 0. Then 4 |f(z)| 6 arctan |z|, for all z ∈ U, π and this inequality is sharp for each point z ∈ U. In 1977, H. W. Hethcote [11] improved this result by removing the assumption f(0) = 0 and proved the following theorem. Theorem 2 ([11, Theorem 1] and [26, Theorem 3.6.1]). Let f : U → U be a harmonic function. Then 1 − |z|2 4 f(z) − f(0) 6 arctan |z|, for all z ∈ U. 1+ |z|2 π As it was written in [23], it seems that researchers have had some difficulties to arXiv:1911.03589v1 [math.CV] 9 Nov 2019 handle the case f(0) 6= 0, where f is harmonic mapping from U to itself. Before explaining the essence of these difficulties, it is necessary to recall one mapping and some of its properties.
    [Show full text]
  • Algebraicity Criteria and Their Applications
    Algebraicity Criteria and Their Applications The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Tang, Yunqing. 2016. Algebraicity Criteria and Their Applications. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493480 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Algebraicity criteria and their applications A dissertation presented by Yunqing Tang to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts May 2016 c 2016 – Yunqing Tang All rights reserved. DissertationAdvisor:ProfessorMarkKisin YunqingTang Algebraicity criteria and their applications Abstract We use generalizations of the Borel–Dwork criterion to prove variants of the Grothedieck–Katz p-curvature conjecture and the conjecture of Ogus for some classes of abelian varieties over number fields. The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for all but finitely many primes p,hasfinite monodromy. It is known that it suffices to prove the conjecture for differential equations on P1 − 0, 1, . We prove a variant of this conjecture for P1 0, 1, , which asserts that if the equation { ∞} −{ ∞} satisfies a certain convergence condition for all p, then its monodromy is trivial.
    [Show full text]
  • Applications of the Cauchy Theory
    Chapter 4 Applications Of The Cauchy Theory This chapter contains several applications of the material developed in Chapter 3. In the first section, we will describe the possible behavior of an analytic function near a singularity of that function. 4.1 Singularities We will say that f has an isolated singularity at z0 if f is analytic on D(z0,r) \{z0} for some r. What, if anything, can be said about the behavior of f near z0? The basic tool needed to answer this question is the Laurent series, an expansion of f(z)in powers of z − z0 in which negative as well as positive powers of z − z0 may appear. In fact, the number of negative powers in this expansion is the key to determining how f behaves near z0. From now on, the punctured disk D(z0,r) \{z0} will be denoted by D (z0,r). We will need a consequence of Cauchy’s integral formula. 4.1.1 Theorem Let f be analytic on an open set Ω containing the annulus {z : r1 ≤|z − z0|≤r2}, 0 <r1 <r2 < ∞, and let γ1 and γ2 denote the positively oriented inner and outer boundaries of the annulus. Then for r1 < |z − z0| <r2, we have 1 f(w) − 1 f(w) f(z)= − dw − dw. 2πi γ2 w z 2πi γ1 w z Proof. Apply Cauchy’s integral formula [part (ii)of (3.3.1)]to the cycle γ2 − γ1. ♣ 1 2 CHAPTER 4. APPLICATIONS OF THE CAUCHY THEORY 4.1.2 Definition For 0 ≤ s1 <s2 ≤ +∞ and z0 ∈ C, we will denote the open annulus {z : s1 < |z−z0| <s2} by A(z0,s1,s2).
    [Show full text]
  • Lectures on Non-Archimedean Function Theory Advanced
    Lectures on Non-Archimedean Function Theory Advanced School on p-Adic Analysis and Applications The Abdus Salam International Centre for Theoretical Physics Trieste, Italy William Cherry Department of Mathematics University of North Texas August 31 – September 4, 2009 Lecture 1: Analogs of Basic Complex Function Theory Lecture 2: Valuation Polygons and a Poisson-Jensen Formula Lecture 3: Non-Archimedean Value Distribution Theory Lecture 4: Benedetto’s Non-Archimedean Island Theorems arXiv:0909.4509v1 [math.CV] 24 Sep 2009 Non-Archimedean Function Theory: Analogs of Basic Complex Function Theory 3 This lecture series is an introduction to non-Archimedean function theory. The audience is assumed to be familiar with non-Archimedean fields and non-Archimedean absolute values, as well as to have had a standard introductory course in complex function theory. A standard reference for the later could be, for example, [Ah 2]. No prior exposure to non-Archimedean function theory is supposed. Full details on the basics of non-Archimedean absolute values and the construction of p-adic number fields, the most important of the non-Archimedean fields, can be found in [Rob]. 1 Analogs of Basic Complex Function Theory 1.1 Non-Archimedean Fields Let A be a commutative ring. A non-Archimedean absolute value | | on A is a function from A to the non-negative real numbers R≥0 satisfying the following three properties: AV 1. |a| = 0 if and only if a = 0; AV 2. |ab| = |a| · |b| for all a, b ∈ A; and AV 3. |a + b|≤ max{|a|, |b|} for all a, b ∈ A. Exercise 1.1.1.
    [Show full text]
  • Riemann's Mapping Theorem
    4. Del Riemann’s mapping theorem version 0:21 | Tuesday, October 25, 2016 6:47:46 PM very preliminary version| more under way. One dares say that the Riemann mapping theorem is one of more famous theorem in the whole science of mathematics. Together with its generalization to Riemann surfaces, the so called Uniformisation Theorem, it is with out doubt the corner-stone of function theory. The theorem classifies all simply connected Riemann-surfaces uo to biholomopisms; and list is astonishingly short. There are just three: The unit disk D, the complex plane C and the Riemann sphere C^! Riemann announced the mapping theorem in his inaugural dissertation1 which he defended in G¨ottingenin . His version a was weaker than full version of today, in that he seems only to treat domains bounded by piecewise differentiable Jordan curves. His proof was not waterproof either, lacking arguments for why the Dirichlet problem has solutions. The fault was later repaired by several people, so his method is sound (of course!). In the modern version there is no further restrictions on the domain than being simply connected. William Fogg Osgood was the first to give a complete proof of the theorem in that form (in ), but improvements of the proof continued to come during the first quarter of the th century. We present Carath´eodory's version of the proof by Lip´otFej´erand Frigyes Riesz, like Reinholdt Remmert does in his book [?], and we shall closely follow the presentation there. This chapter starts with the legendary lemma of Schwarz' and a study of the biho- lomorphic automorphisms of the unit disk.
    [Show full text]
  • Schwarz-Christoffel Transformations by Philip P. Bergonio (Under the Direction of Edward Azoff) Abstract the Riemann Mapping
    Schwarz-Christoffel Transformations by Philip P. Bergonio (Under the direction of Edward Azoff) Abstract The Riemann Mapping Theorem guarantees that the upper half plane is conformally equivalent to the interior domain determined by any polygon. Schwarz-Christoffel transfor- mations provide explicit formulas for the maps that work. Popular textbook treatments of the topic range from motivational and contructive to proof-oriented. The aim of this paper is to combine the strengths of these expositions, filling in details and adding more information when necessary. In particular, careful attention is paid to the crucial fact, taken for granted in most elementary texts, that all conformal equivalences between the domains in question extend continuously to their closures. Index words: Complex Analysis, Schwarz-Christoffel Transformations, Polygons in the Complex Plane Schwarz-Christoffel Transformations by Philip P. Bergonio B.S., Georgia Southwestern State University, 2003 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Master of Arts Athens, Georgia 2007 c 2007 Philip P. Bergonio All Rights Reserved Schwarz-Christoffel Transformations by Philip P. Bergonio Approved: Major Professor: Edward Azoff Committee: Daniel Nakano Shuzhou Wang Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia December 2007 Table of Contents Page Chapter 1 Introduction . 1 2 Background Information . 5 2.1 Preliminaries . 5 2.2 Linear Curves and Polygons . 8 3 Two Examples and Motivation for the Formula . 13 3.1 Prototypical Examples . 13 3.2 Motivation for the Formula . 16 4 Properties of Schwarz-Christoffel Candidates . 19 4.1 Well-Definedness of f .....................
    [Show full text]
  • A Note on the Schwarz Lemma for Harmonic Functions
    Filomat 34:11 (2020), 3711–3720 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL2011711S University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Note on the Schwarz Lemma for Harmonic Functions Marek Svetlika aUniversity of Belgrade, Faculty of Mathematics Abstract. In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z 0 are given. 1. Introduction 1.1. A summary of some results In this paper we consider some generalizations of the Schwarz lemma for harmonic functions from the unit disk U tz P C : |z| 1u to the interval p¡1; 1q (or to itself). First, we cite a theorem which is known as the Schwarz lemma for harmonic functions and is considered a classical result. Theorem 1 ([10],[9, p.77]). Let f : U Ñ U be a harmonic function such that f p0q 0. Then 4 | f pzq| ¤ arctan |z|; for all z P U; π and this inequality is sharp for each point z P U. In 1977, H. W. Hethcote [11] improved this result by removing the assumption f p0q 0 and proved the following theorem. Theorem 2 ([11, Theorem 1] and [29, Theorem 3.6.1]). Let f : U Ñ U be a harmonic function. Then § § § 2 § § 1 ¡ |z| § 4 § f pzq ¡ f p0q§ ¤ arctan |z|; for all z P U: 1 |z|2 π As was written in [25], it seems that researchers had some difficulties handling the case f p0q 0, where f is a harmonic mapping from U to itself.
    [Show full text]
  • 1. a Maximum Modulus Principle for Analytic Polynomials in the Following Problems, We Outline Two Proofs of a Version of Maximum Mod- Ulus Principle
    PROBLEMS IN COMPLEX ANALYSIS SAMEER CHAVAN 1. A Maximum Modulus Principle for Analytic Polynomials In the following problems, we outline two proofs of a version of Maximum Mod- ulus Principle. The first one is based on linear algebra (not the simplest one). Problem 1.1 (Orr Morshe Shalit, Amer. Math. Monthly). Let p(z) = a0 + a1z + n p 2 ··· + anz be an analytic polynomial and let s := 1 − jzj for z 2 C with jzj ≤ 1. Let ei denote the column n × 1 matrix with 1 at the ith place and 0 else. Verify: (1) Consider the (n+1)×(n+1) matrix U with columns ze1+se2; e3; e4; ··· ; en+1, and se1 − ze¯ 2 (in order). Then U is unitary with eigenvalues λ1; ··· ; λn+1 of modulus 1 (Hint. Check that columns of U are mutually orthonormal). k t k (2) z = (e1) U e1 (Check: Apply induction on k), and hence t p(z) = (e1) p(U)e1: (3) maxjz|≤1 jp(z)j ≤ kp(U)k (Hint. Recall that kABk ≤ kAkkBk) (4) If D is the diagonal matrix with diagonal entries λ1; ··· ; λn+1 then kp(U)k = kp(D)k = max jp(λi)j: i=1;··· ;n+1 Conclude that maxjz|≤1 jp(z)j = maxjzj=1 jp(z)j. Problem 1.2 (Walter Rudin, Real and Complex Analysis). Let p(z) = a0 + a1z + n ··· + anz be an analytic polynomial. Let z0 2 C be such that jf(z)j ≤ jf(z0)j: n Assume jz0j < 1; and write p(z) = b0+b1(z−z0)+···+bn(z−z0) .
    [Show full text]
  • Chapter 2 Elementary Properties of Holomorphic Functions
    Chapter 2 Elementary properties of holomorphic functions in several variables NOT original material NOT intended for distribution §2.1: Holomorphy for functions of several variables §2.2: The Cauchy formula for polydiscs and its elementary consequences §2.3: Hartogs’ Theorem on separately holomorphic functions §2.4: Solving the ∂–equation in a polydisc and holomorphic extension §2.5: Local solution of the ∂–equation for p, q–forms §2.6: Power series and Reinhardt domains §2.7: Domains of holomorphy and holomorphic convexity §2.8: Pseudoconvexity, the ball versus the polydisc §2.9: CR-structures and the Lewy extension theorem §2.10: The Weierstraß preparation theorem 2.1 Holomorphy for functions of several variables In this chapter we introduce holomorphic functions of several variables and deduce their simpler prop- erties. Much is routine generalization from the one–variable case via the Cauchy integral formula. Though even the elementary theory of the ∂–equation is more involved. The extension theorems in several variables are quite different from the single variable case; there is a straightforward analogue of Riemann’s removable singularities but Hartogs’ theorem is truly a multi–variable result. We consider the theory of power series in many variables, their convergence properties are quite different from the single variable case. This follows in part from the Hartogs’ result mentioned above. To begin to understand these new phenomena we consider various notions of ‘convexity’. We begin with the real vector space Rm. An automorphism, J of T Rm, that may depend smoothly on the point and satisfies (2.1.1) J 2 = − Id is called an almost complex structure.
    [Show full text]
  • RIGIDITY of HOLOMORPHIC MAPPINGS and a NEW SCHWARZ LEMMA at Me BOUNDARY
    JOURNAL OF 1lfE AMERICAN MAlHEMATICAL SOCIETY Volume 7, Number 3, July 1994 RIGIDITY OF HOLOMORPHIC MAPPINGS AND A NEW SCHWARZ LEMMA AT mE BOUNDARY DANIEL M. BURNS AND STEVEN G. KRA..."'ITZ 1. INTRODUCTION The uniqueness portion of the classical Schwarz lemma has played an inter- esting historical role. Usually first encountered in the classification of the con- formal self-maps of the disc, it arises in more general considerations of automor- phism groups, in the construction of the Caratheodory and Kobayashi/Royden metrics, and in a variety of contexts in analysis on manifolds. Useful references for this material are [AHL, KRl, KOB, GK1, GK2, YAU]. Generalizations of the disc Schwarz lemma to multiply-connected domains in one complex variable are generated naturally and easily using the uniformization theorem. The generalization to several complex variables requires insights of a different nature. For example, the so-called Caratheodory-Cartan-Kaup-Wu theorem [WU] says in part that if Q is a bounded domain in en and if CI> is a holomorphic self-map of Q that fixes a point P E Q, then the holomorphic Jacobian determinant det JacCl>(P) has modulus less than or equal to 1, and equals 1 if and only if CI> is a biholomorphism of Q. An interesting and important feature of this result is that it has a global hypothesis (that Q be mapped to itself) and a local hypothesis (the condition on the behavior of the mapping at P). The conclusion is then a strong global one. We will see this paradigm repeated in the work that follows.
    [Show full text]
  • Riemann Mapping Theorem
    RIEMANN MAPPING THEOREM VED V. DATAR∗ Recall that two domains are called conformally equivalent if there exists a holomorphic bijection from one to the other. This automatically implies that there is an inverse holomorphic function. The aim of this lecture is to prove the following deep theorem due to Riemann. Denote by D the unit disc centered at the origin. Theorem 0.1. Let Ω ⊂ C be a simply connected set that is not all of C. Then for any z0 2 Ω, there exists a unique biholomorphism F :Ω ! D such that 0 F (z0) = 0; and F (z0) > 0: 0 0 Here F (z0) > 0 stands for F (z0) being real and positive. Note that by Liouville's theorem, such a statement is patently false if Ω = C, and so the hypothesis that Ω is a proper subset is a necessary condition. As a consequence of the Theorem, we have the following corollary. Corollary 0.1. Any two proper, simply connected subsets for C are confor- mally equivalent. Uniqueness The uniqueness follows easily from the following lemma. Lemma 0.1 (Schwarz's lemma). Let f : D ! D be a holomorphic map such that f(0) = 0. Then jf(z)j < jzj 0 for all z 2 D. Moreover, if jf(z)j = jzj for some non-zero z, or jf (0)j = 1, then f(z) = az for some a 2 C with jaj = 1. Proof. Consider the holomorphic function on D n f0g defined by f(z) g(z) := : z Then since f(0) = 0, g(z) has a removable singularity at z = 0, and hence extends to a holomorphic function on all of D.
    [Show full text]